This course provides an introduction to the Atiyah-Singer index theorem, with a particular focus on its local version and applications in geometry and topology. The Atiyah-Singer index theorem is one of the cornerstones of modern mathematics, connecting analysis, geometry, and topology in profound ways. This course aims to develop an understanding of the analytical and geometrical foundations of the theorem and its implications in various mathematical contexts.
The course will cover:
Background on elliptic operators and their properties.
Basic concepts of differential geometry relevant to index theory, including vector bundles, connections, and curvature.
The heat kernel approach and its role in the local index theorem.
Key ideas and proof sketches of the Atiyah-Singer index theorem.
Applications of the local index theorem in geometry and topology, such as the Hirzebruch signature theorem and the Gauss-Bonnet theorem.
By the end of the course, students will have a conceptual understanding of the Atiyah-Singer index theorem, its local formulation, and the interplay between analysis, geometry, and topology.
Basic knowledge of differential geometry (manifolds, tangent spaces, vector bundles, connections).
Familiarity with functional analysis (Hilbert spaces, bounded operators) and partial differential equations (Laplacians).
Some exposure to topology (homology, cohomology) is recommended.
Heat kernels and Dirac operators by N. Berline, E. Getzler, and M. Vergne
A comprehensive introduction to the heat kernel approach to the Atiyah-Singer index theorem.
Elliptic operators, topology, and asymptotic methods by J. Roe
An accessible introduction to elliptic operators and index theory.
Topology and analysis: the Atiyah-Singer index formula and gauge-theoretic physics by B. Booss and D. D. Bleecker
A detailed exploration of the index formula and its applications in mathematical physics, translated from German by Bleecker and A. Mader.
Spin geometry by H. B. Lawson and M.-L. Michelsohn
Provides background on spin geometry and its relationship to the index theorem.
Differential forms in algebraic topology by R. Bott and L. W. Tu
Offers foundational material in differential forms and their applications in topology.